Tree-based models
1 Loading the data and the packages
First, the packages
require("CASdatasets")
require("rpart")
require("rpart.plot")
require("caret")then, the data
# data("freMTPLfreq")
# freMTPLfreq = subset(freMTPLfreq, Exposure<=1 & Exposure >= 0 & CarAge<=25)
#
# set.seed(85)
# folds = createDataPartition(freMTPLfreq$ClaimNb, 0.5)
# dataset = freMTPLfreq[folds[[1]], ]
load("dataset.RData")Let us first split out dataset in two parts: a training set and a testing set.
set.seed(21)
in_training = createDataPartition(dataset$ClaimNb, times = 1, p = 0.8, list=FALSE)
training_set = dataset[in_training,]
testing_set = dataset[-in_training,]2 CART
The package rpart allows to compute regression trees. rpart can be used for regression and classification. It also implements a method for poisson data.
2.1 Example
Let us start with a simple example:
m0_rpart = rpart(cbind(Exposure, ClaimNb)~ DriverAge + CarAge,
data=training_set,
method="poisson")
summary(m0_rpart)## Call:
## rpart(formula = cbind(Exposure, ClaimNb) ~ DriverAge + CarAge,
## data = training_set, method = "poisson")
## n= 164346
##
## CP nsplit rel error xerror xstd
## 1 0.007291066 0 1 0 0
##
## Node number 1: 164346 observations
## events=6440, estimated rate=0.06993893 , mean deviance=0.2560171It appears that the tree has a single node and has not been split further. This comes from the complexity parameter which penalizes the splitting. By default, the complexity parameter \(cp\) is set to \(0.01\), which is often too large for Poisson data with low frequencies.
Let us put \(cp = 0\), but to keep a small tree we will also impose a maximum depth of \(3\).
m0_rpart = rpart(cbind(Exposure, ClaimNb)~ DriverAge + CarAge,
data=training_set,
method="poisson",
control = rpart.control(cp = 0, maxdepth = 3))
summary(m0_rpart)## Call:
## rpart(formula = cbind(Exposure, ClaimNb) ~ DriverAge + CarAge,
## data = training_set, method = "poisson", control = rpart.control(cp = 0,
## maxdepth = 3))
## n= 164346
##
## CP nsplit rel error xerror xstd
## 1 0.0072910657 0 1.0000000 1.0000947 0.009761534
## 2 0.0017755017 1 0.9927089 0.9939987 0.009678228
## 3 0.0008692125 2 0.9909334 0.9929020 0.009672146
## 4 0.0005163174 3 0.9900642 0.9909126 0.009652092
## 5 0.0004932740 4 0.9895479 0.9908982 0.009651163
## 6 0.0002693544 5 0.9890546 0.9905951 0.009648972
## 7 0.0002409219 6 0.9887853 0.9907275 0.009649443
## 8 0.0000000000 7 0.9885444 0.9908074 0.009651133
##
## Variable importance
## DriverAge CarAge
## 88 12
##
## Node number 1: 164346 observations, complexity param=0.007291066
## events=6440, estimated rate=0.06993893 , mean deviance=0.2560171
## left son=2 (151464 obs) right son=3 (12882 obs)
## Primary splits:
## DriverAge < 26.5 to the right, improve=306.77560, (0 missing)
## CarAge < 17.5 to the right, improve= 44.55588, (0 missing)
##
## Node number 2: 151464 observations, complexity param=0.0008692125
## events=5671, estimated rate=0.0655778 , mean deviance=0.2453646
## left son=4 (5926 obs) right son=5 (145538 obs)
## Primary splits:
## CarAge < 17.5 to the right, improve=36.57344, (0 missing)
## DriverAge < 53.5 to the right, improve=23.06150, (0 missing)
##
## Node number 3: 12882 observations, complexity param=0.001775502
## events=769, estimated rate=0.1371023 , mean deviance=0.3574526
## left son=6 (9846 obs) right son=7 (3036 obs)
## Primary splits:
## DriverAge < 21.5 to the right, improve=74.72201, (0 missing)
## CarAge < 14.5 to the right, improve=14.40992, (0 missing)
##
## Node number 4: 5926 observations, complexity param=0.0002693544
## events=156, estimated rate=0.04232996 , mean deviance=0.1952838
## left son=8 (423 obs) right son=9 (5503 obs)
## Primary splits:
## DriverAge < 72.5 to the right, improve=11.474520, (0 missing)
## CarAge < 23.5 to the left, improve= 1.969211, (0 missing)
##
## Node number 5: 145538 observations, complexity param=0.000493274
## events=5515, estimated rate=0.06661995 , mean deviance=0.2471525
## left son=10 (42343 obs) right son=11 (103195 obs)
## Primary splits:
## DriverAge < 53.5 to the right, improve=20.75470, (0 missing)
## CarAge < 12.5 to the right, improve=13.35384, (0 missing)
##
## Node number 6: 9846 observations, complexity param=0.0005163174
## events=498, estimated rate=0.1132733 , mean deviance=0.3182285
## left son=12 (1179 obs) right son=13 (8667 obs)
## Primary splits:
## CarAge < 14.5 to the right, improve=21.725950, (0 missing)
## DriverAge < 23.5 to the right, improve= 7.424345, (0 missing)
##
## Node number 7: 3036 observations, complexity param=0.0002409219
## events=271, estimated rate=0.2219925 , mean deviance=0.4600535
## left son=14 (2188 obs) right son=15 (848 obs)
## Primary splits:
## DriverAge < 19.5 to the right, improve=10.246780, (0 missing)
## CarAge < 5.5 to the left, improve= 3.551826, (0 missing)
##
## Node number 8: 423 observations
## events=4, estimated rate=0.01396801 , mean deviance=0.08591919
##
## Node number 9: 5503 observations
## events=152, estimated rate=0.04546408 , mean deviance=0.2016309
##
## Node number 10: 42343 observations
## events=1652, estimated rate=0.06082192 , mean deviance=0.2540509
##
## Node number 11: 103195 observations
## events=3863, estimated rate=0.06945315 , mean deviance=0.2441208
##
## Node number 12: 1179 observations
## events=30, estimated rate=0.05602265 , mean deviance=0.193547
##
## Node number 13: 8667 observations
## events=468, estimated rate=0.121307 , mean deviance=0.3326828
##
## Node number 14: 2188 observations
## events=178, estimated rate=0.1954661 , mean deviance=0.4254648
##
## Node number 15: 848 observations
## events=93, estimated rate=0.2902984 , mean deviance=0.5373451The easiest way to interpret a CART is probably to plot it (if it is not too large, though!). This can be achieved with the function rpart.plot from the package rpart.plot.
rpart.plot(m0_rpart)
If the tree is too large, we will probably have some overfitting. To prevent overfitting, we can play with the complexity parameter cp. A good approach is to compute the whole tree, without any penalty (i.e. complexity parameter is set to 0) and afterwards prune to tree.
m0_rpart = rpart(cbind(Exposure, ClaimNb)~ DriverAge + CarAge,
data=training_set,
method="poisson",
control = rpart.control(cp = 0))
rpart.plot(m0_rpart)## Warning: labs do not fit even at cex 0.15, there may be some overplotting
The trees becomes too large. We can prune the tree using the function prune. For instance, if we set \(cp = 0.0009\),
rpart.plot(prune(m0_rpart, cp=0.0009))
We also see that in some terminal nodes (i.e. leaves), the number of observations (and of claims) is very low. We can set a minimum number of observation in any terminal node using minbucket
m0_rpart = rpart(cbind(Exposure, ClaimNb)~ DriverAge + CarAge,
data=training_set,
method="poisson",
control = rpart.control(cp = 0, maxdepth = 3, minbucket = 1000))
rpart.plot(m0_rpart)
2.2 Cross-Validation
Let us now find the optimal tree, by using cross-validation. We will again only use the variable DriverAge and CarAge in this section. By default, rpart will perform 10-fold cross-validation, using the option xval = 10.
m0_rpart = rpart(cbind(Exposure, ClaimNb)~ DriverAge + CarAge,
data=training_set,
method="poisson",
control = rpart.control(cp = 0, xval = 10))
printcp(m0_rpart)##
## Rates regression tree:
## rpart(formula = cbind(Exposure, ClaimNb) ~ DriverAge + CarAge,
## data = training_set, method = "poisson", control = rpart.control(cp = 0,
## xval = 10))
##
## Variables actually used in tree construction:
## [1] CarAge DriverAge
##
## Root node error: 42075/164346 = 0.25602
##
## n= 164346
##
## CP nsplit rel error xerror xstd
## 1 7.2911e-03 0 1.00000 1.00003 0.0097610
## 2 1.7755e-03 1 0.99271 0.99402 0.0096787
## 3 8.6921e-04 2 0.99093 0.99263 0.0096691
## 4 5.1632e-04 3 0.99006 0.99033 0.0096442
## 5 4.9327e-04 4 0.98955 0.99041 0.0096450
## 6 2.9455e-04 5 0.98905 0.99030 0.0096419
## 7 2.7243e-04 6 0.98876 0.98999 0.0096401
## 8 2.6935e-04 7 0.98849 0.98999 0.0096402
## 9 2.6520e-04 8 0.98822 0.99000 0.0096397
## 10 2.4092e-04 9 0.98795 0.98991 0.0096392
## 11 1.9846e-04 10 0.98771 0.98973 0.0096373
## 12 1.9348e-04 11 0.98751 0.98973 0.0096380
## 13 1.7816e-04 12 0.98732 0.98988 0.0096446
## 14 1.7661e-04 13 0.98714 0.99007 0.0096489
## 15 1.7231e-04 15 0.98679 0.99024 0.0096513
## 16 1.5411e-04 17 0.98644 0.99037 0.0096527
## 17 1.2138e-04 19 0.98614 0.99102 0.0096594
## 18 1.2052e-04 20 0.98601 0.99165 0.0096753
## 19 1.1963e-04 23 0.98565 0.99161 0.0096748
## 20 1.1704e-04 25 0.98541 0.99168 0.0096764
## 21 1.1392e-04 27 0.98518 0.99191 0.0096818
## 22 1.1297e-04 29 0.98495 0.99198 0.0096844
## 23 1.0920e-04 33 0.98450 0.99210 0.0096866
## 24 1.0034e-04 36 0.98417 0.99257 0.0096952
## 25 9.8327e-05 38 0.98397 0.99331 0.0097072
## 26 9.6339e-05 42 0.98356 0.99372 0.0097154
## 27 9.5145e-05 46 0.98317 0.99402 0.0097219
## 28 9.4845e-05 48 0.98298 0.99421 0.0097251
## 29 9.4760e-05 49 0.98289 0.99413 0.0097240
## 30 8.8055e-05 50 0.98279 0.99479 0.0097330
## 31 8.4263e-05 51 0.98271 0.99525 0.0097517
## 32 8.2453e-05 55 0.98237 0.99622 0.0097634
## 33 8.2064e-05 56 0.98229 0.99648 0.0097758
## 34 8.1953e-05 57 0.98220 0.99649 0.0097760
## 35 8.1321e-05 58 0.98212 0.99645 0.0097754
## 36 8.1028e-05 59 0.98204 0.99637 0.0097735
## 37 7.9678e-05 62 0.98180 0.99646 0.0097755
## 38 7.9014e-05 64 0.98164 0.99694 0.0097846
## 39 7.9005e-05 65 0.98156 0.99704 0.0097849
## 40 7.8608e-05 75 0.98074 0.99700 0.0097839
## 41 7.8565e-05 77 0.98059 0.99708 0.0097880
## 42 7.7100e-05 78 0.98051 0.99709 0.0097899
## 43 7.6702e-05 79 0.98043 0.99736 0.0097931
## 44 7.6583e-05 81 0.98028 0.99748 0.0097954
## 45 7.4991e-05 87 0.97981 0.99806 0.0098039
## 46 7.4968e-05 91 0.97950 0.99811 0.0098042
## 47 7.3551e-05 93 0.97936 0.99816 0.0098070
## 48 7.2275e-05 95 0.97921 0.99861 0.0098140
## 49 7.2046e-05 98 0.97899 0.99899 0.0098243
## 50 7.2000e-05 99 0.97892 0.99913 0.0098269
## 51 7.1303e-05 101 0.97878 0.99924 0.0098298
## 52 7.0581e-05 102 0.97870 0.99958 0.0098372
## 53 6.9908e-05 104 0.97856 0.99973 0.0098396
## 54 6.9112e-05 110 0.97814 0.99982 0.0098392
## 55 6.8240e-05 123 0.97721 1.00034 0.0098456
## 56 6.7850e-05 138 0.97607 1.00039 0.0098463
## 57 6.7378e-05 140 0.97593 1.00058 0.0098487
## 58 6.6852e-05 141 0.97586 1.00074 0.0098513
## 59 6.5990e-05 142 0.97580 1.00105 0.0098559
## 60 6.4397e-05 143 0.97573 1.00162 0.0098633
## 61 6.2781e-05 144 0.97567 1.00205 0.0098665
## 62 6.2351e-05 146 0.97554 1.00207 0.0098675
## 63 6.2257e-05 150 0.97529 1.00203 0.0098661
## 64 6.2075e-05 151 0.97523 1.00218 0.0098676
## 65 6.1982e-05 154 0.97504 1.00215 0.0098676
## 66 6.1787e-05 158 0.97480 1.00227 0.0098693
## 67 6.1241e-05 162 0.97455 1.00231 0.0098701
## 68 6.0337e-05 163 0.97449 1.00235 0.0098706
## 69 5.9564e-05 164 0.97443 1.00297 0.0098819
## 70 5.9397e-05 166 0.97431 1.00319 0.0098847
## 71 5.9046e-05 168 0.97419 1.00329 0.0098858
## 72 5.8231e-05 169 0.97413 1.00356 0.0098895
## 73 5.8173e-05 172 0.97396 1.00370 0.0098927
## 74 5.7809e-05 173 0.97390 1.00371 0.0098930
## 75 5.7589e-05 176 0.97372 1.00375 0.0098937
## 76 5.6754e-05 177 0.97367 1.00402 0.0098959
## 77 5.6492e-05 178 0.97361 1.00425 0.0098994
## 78 5.6212e-05 180 0.97350 1.00438 0.0099017
## 79 5.5075e-05 183 0.97333 1.00455 0.0099063
## 80 5.4844e-05 185 0.97322 1.00531 0.0099169
## 81 5.4384e-05 186 0.97316 1.00529 0.0099172
## 82 5.2792e-05 187 0.97311 1.00552 0.0099214
## 83 5.2347e-05 189 0.97300 1.00609 0.0099295
## 84 5.2256e-05 190 0.97295 1.00622 0.0099306
## 85 5.2207e-05 193 0.97279 1.00640 0.0099329
## 86 5.2061e-05 200 0.97242 1.00645 0.0099336
## 87 5.1513e-05 207 0.97206 1.00661 0.0099375
## 88 5.0497e-05 214 0.97170 1.00689 0.0099409
## 89 5.0462e-05 216 0.97160 1.00699 0.0099429
## 90 5.0440e-05 219 0.97144 1.00699 0.0099429
## 91 5.0341e-05 228 0.97098 1.00717 0.0099439
## 92 5.0307e-05 232 0.97078 1.00719 0.0099462
## 93 5.0244e-05 233 0.97073 1.00719 0.0099462
## 94 4.9958e-05 242 0.97024 1.00713 0.0099456
## 95 4.9786e-05 243 0.97019 1.00715 0.0099461
## 96 4.9469e-05 246 0.97004 1.00717 0.0099472
## 97 4.9124e-05 253 0.96967 1.00719 0.0099490
## 98 4.9063e-05 258 0.96943 1.00717 0.0099489
## 99 4.9018e-05 260 0.96933 1.00718 0.0099491
## 100 4.8514e-05 261 0.96928 1.00718 0.0099498
## 101 4.8456e-05 265 0.96909 1.00735 0.0099541
## 102 4.8383e-05 266 0.96904 1.00757 0.0099574
## 103 4.8077e-05 268 0.96894 1.00755 0.0099576
## 104 4.8012e-05 269 0.96889 1.00757 0.0099576
## 105 4.7318e-05 271 0.96880 1.00752 0.0099573
## 106 4.7091e-05 272 0.96875 1.00779 0.0099612
## 107 4.6972e-05 275 0.96861 1.00785 0.0099623
## 108 4.6497e-05 278 0.96847 1.00767 0.0099612
## 109 4.4848e-05 279 0.96842 1.00838 0.0099714
## 110 4.4691e-05 282 0.96829 1.00841 0.0099714
## 111 4.4605e-05 283 0.96824 1.00843 0.0099726
## 112 4.3967e-05 292 0.96781 1.00834 0.0099711
## 113 4.3612e-05 296 0.96763 1.00827 0.0099702
## 114 4.3529e-05 298 0.96754 1.00835 0.0099705
## 115 4.3523e-05 300 0.96746 1.00858 0.0099754
## 116 4.3327e-05 302 0.96737 1.00857 0.0099749
## 117 4.3081e-05 303 0.96732 1.00857 0.0099748
## 118 4.3003e-05 306 0.96720 1.00858 0.0099753
## 119 4.2460e-05 307 0.96715 1.00860 0.0099754
## 120 4.2144e-05 310 0.96703 1.00872 0.0099810
## 121 4.1561e-05 312 0.96694 1.00865 0.0099815
## 122 4.1341e-05 313 0.96690 1.00883 0.0099827
## 123 4.1239e-05 314 0.96686 1.00882 0.0099827
## 124 4.1233e-05 315 0.96682 1.00882 0.0099827
## 125 4.0436e-05 316 0.96678 1.00880 0.0099835
## 126 4.0424e-05 318 0.96669 1.00884 0.0099847
## 127 4.0278e-05 319 0.96665 1.00896 0.0099846
## 128 4.0117e-05 320 0.96661 1.00897 0.0099849
## 129 4.0062e-05 323 0.96649 1.00897 0.0099847
## 130 3.9795e-05 325 0.96641 1.00899 0.0099837
## 131 3.9689e-05 326 0.96637 1.00905 0.0099844
## 132 3.9655e-05 327 0.96633 1.00908 0.0099846
## 133 3.9644e-05 328 0.96629 1.00910 0.0099850
## 134 3.9527e-05 329 0.96625 1.00909 0.0099846
## 135 3.9114e-05 336 0.96597 1.00908 0.0099851
## 136 3.8422e-05 340 0.96581 1.00923 0.0099871
## 137 3.7915e-05 341 0.96577 1.00942 0.0099894
## 138 3.7903e-05 343 0.96569 1.00947 0.0099906
## 139 3.7763e-05 344 0.96566 1.00963 0.0099960
## 140 3.7190e-05 349 0.96546 1.00973 0.0099958
## 141 3.6588e-05 350 0.96542 1.01015 0.0100023
## 142 3.6529e-05 358 0.96512 1.01052 0.0100107
## 143 3.6434e-05 359 0.96508 1.01049 0.0100107
## 144 3.6418e-05 362 0.96497 1.01051 0.0100109
## 145 3.5667e-05 363 0.96493 1.01056 0.0100113
## 146 3.5620e-05 367 0.96479 1.01058 0.0100122
## 147 3.5530e-05 372 0.96460 1.01069 0.0100145
## 148 3.5524e-05 374 0.96453 1.01072 0.0100143
## 149 3.5416e-05 376 0.96446 1.01076 0.0100150
## 150 3.5404e-05 377 0.96442 1.01079 0.0100153
## 151 3.5228e-05 378 0.96439 1.01084 0.0100156
## 152 3.5211e-05 379 0.96435 1.01091 0.0100163
## 153 3.5184e-05 382 0.96424 1.01092 0.0100167
## 154 3.5158e-05 383 0.96421 1.01094 0.0100168
## 155 3.5040e-05 384 0.96417 1.01090 0.0100162
## 156 3.4912e-05 385 0.96414 1.01096 0.0100175
## 157 3.4319e-05 386 0.96410 1.01100 0.0100189
## 158 3.3683e-05 387 0.96407 1.01091 0.0100186
## 159 3.3420e-05 388 0.96404 1.01117 0.0100214
## 160 3.3209e-05 390 0.96397 1.01120 0.0100216
## 161 3.2948e-05 395 0.96380 1.01119 0.0100217
## 162 3.2818e-05 397 0.96374 1.01113 0.0100211
## 163 3.2613e-05 398 0.96371 1.01145 0.0100247
## 164 3.2547e-05 405 0.96348 1.01147 0.0100241
## 165 3.2186e-05 410 0.96331 1.01156 0.0100250
## 166 3.2112e-05 414 0.96318 1.01156 0.0100244
## 167 3.2065e-05 415 0.96314 1.01151 0.0100234
## 168 3.1722e-05 416 0.96311 1.01154 0.0100248
## 169 3.1607e-05 417 0.96308 1.01139 0.0100229
## 170 3.1504e-05 418 0.96305 1.01145 0.0100244
## 171 3.1327e-05 419 0.96302 1.01158 0.0100256
## 172 3.1308e-05 421 0.96295 1.01150 0.0100251
## 173 3.0787e-05 422 0.96292 1.01156 0.0100258
## 174 3.0763e-05 423 0.96289 1.01157 0.0100258
## 175 3.0561e-05 425 0.96283 1.01150 0.0100246
## 176 3.0214e-05 427 0.96277 1.01140 0.0100243
## 177 3.0145e-05 428 0.96274 1.01148 0.0100270
## 178 3.0082e-05 430 0.96268 1.01151 0.0100275
## 179 2.9748e-05 432 0.96262 1.01156 0.0100287
## 180 2.9623e-05 433 0.96259 1.01189 0.0100355
## 181 2.9391e-05 438 0.96244 1.01194 0.0100366
## 182 2.9363e-05 445 0.96222 1.01211 0.0100416
## 183 2.9180e-05 447 0.96216 1.01231 0.0100462
## 184 2.9033e-05 448 0.96213 1.01258 0.0100499
## 185 2.8792e-05 454 0.96196 1.01263 0.0100510
## 186 2.8698e-05 461 0.96175 1.01267 0.0100515
## 187 2.8498e-05 462 0.96172 1.01275 0.0100526
## 188 2.8486e-05 463 0.96169 1.01269 0.0100521
## 189 2.8245e-05 465 0.96163 1.01260 0.0100501
## 190 2.8195e-05 466 0.96160 1.01264 0.0100511
## 191 2.8010e-05 467 0.96158 1.01258 0.0100503
## 192 2.7803e-05 468 0.96155 1.01267 0.0100511
## 193 2.7766e-05 472 0.96144 1.01269 0.0100516
## 194 2.7676e-05 473 0.96141 1.01273 0.0100524
## 195 2.7624e-05 474 0.96138 1.01279 0.0100531
## 196 2.7612e-05 475 0.96135 1.01281 0.0100532
## 197 2.7525e-05 476 0.96133 1.01281 0.0100531
## 198 2.7488e-05 477 0.96130 1.01278 0.0100530
## 199 2.7486e-05 481 0.96118 1.01278 0.0100529
## 200 2.7051e-05 485 0.96107 1.01277 0.0100534
## 201 2.7039e-05 489 0.96096 1.01285 0.0100572
## 202 2.6937e-05 490 0.96094 1.01277 0.0100569
## 203 2.6847e-05 491 0.96091 1.01281 0.0100571
## 204 2.6784e-05 493 0.96085 1.01279 0.0100569
## 205 2.6412e-05 495 0.96080 1.01283 0.0100572
## 206 2.6340e-05 496 0.96077 1.01276 0.0100563
## 207 2.6207e-05 497 0.96075 1.01280 0.0100563
## 208 2.6168e-05 498 0.96072 1.01288 0.0100586
## 209 2.6065e-05 499 0.96070 1.01292 0.0100588
## 210 2.5956e-05 501 0.96064 1.01298 0.0100596
## 211 2.5771e-05 502 0.96062 1.01300 0.0100585
## 212 2.5665e-05 503 0.96059 1.01300 0.0100588
## 213 2.5495e-05 505 0.96054 1.01306 0.0100596
## 214 2.5372e-05 507 0.96049 1.01308 0.0100597
## 215 2.5346e-05 508 0.96046 1.01309 0.0100598
## 216 2.5040e-05 509 0.96044 1.01307 0.0100606
## 217 2.4811e-05 510 0.96041 1.01312 0.0100609
## 218 2.4712e-05 511 0.96039 1.01315 0.0100616
## 219 2.4681e-05 514 0.96032 1.01322 0.0100623
## 220 2.4649e-05 516 0.96027 1.01321 0.0100621
## 221 2.4599e-05 517 0.96024 1.01313 0.0100613
## 222 2.4400e-05 519 0.96019 1.01320 0.0100619
## 223 2.4248e-05 520 0.96017 1.01327 0.0100625
## 224 2.3806e-05 524 0.96007 1.01331 0.0100629
## 225 2.3748e-05 527 0.96000 1.01336 0.0100629
## 226 2.3636e-05 528 0.95997 1.01321 0.0100612
## 227 2.3627e-05 530 0.95992 1.01322 0.0100612
## 228 2.3614e-05 532 0.95988 1.01322 0.0100612
## 229 2.3566e-05 533 0.95985 1.01323 0.0100614
## 230 2.3226e-05 539 0.95969 1.01329 0.0100623
## 231 2.3098e-05 540 0.95966 1.01325 0.0100614
## 232 2.2615e-05 542 0.95962 1.01336 0.0100618
## 233 2.2562e-05 545 0.95955 1.01341 0.0100620
## 234 2.2410e-05 546 0.95953 1.01339 0.0100623
## 235 2.2228e-05 547 0.95950 1.01339 0.0100623
## 236 2.2199e-05 548 0.95948 1.01345 0.0100630
## 237 2.2168e-05 551 0.95941 1.01349 0.0100634
## 238 2.1965e-05 552 0.95939 1.01353 0.0100637
## 239 2.1785e-05 553 0.95937 1.01359 0.0100651
## 240 2.1772e-05 557 0.95927 1.01364 0.0100657
## 241 2.1673e-05 558 0.95925 1.01365 0.0100657
## 242 2.1021e-05 560 0.95921 1.01368 0.0100653
## 243 2.0929e-05 562 0.95917 1.01386 0.0100680
## 244 2.0898e-05 563 0.95915 1.01393 0.0100690
## 245 2.0687e-05 564 0.95913 1.01400 0.0100695
## 246 2.0600e-05 567 0.95906 1.01400 0.0100696
## 247 2.0511e-05 568 0.95904 1.01401 0.0100697
## 248 2.0496e-05 569 0.95902 1.01400 0.0100697
## 249 2.0409e-05 570 0.95900 1.01401 0.0100697
## 250 2.0380e-05 571 0.95898 1.01403 0.0100699
## 251 2.0021e-05 572 0.95895 1.01412 0.0100712
## 252 1.9892e-05 573 0.95893 1.01427 0.0100737
## 253 1.9826e-05 574 0.95891 1.01426 0.0100728
## 254 1.9710e-05 576 0.95888 1.01429 0.0100731
## 255 1.9691e-05 577 0.95886 1.01426 0.0100728
## 256 1.9607e-05 580 0.95880 1.01423 0.0100719
## 257 1.9519e-05 581 0.95878 1.01417 0.0100713
## 258 1.9320e-05 582 0.95876 1.01419 0.0100717
## 259 1.9243e-05 583 0.95874 1.01413 0.0100714
## 260 1.9192e-05 584 0.95872 1.01411 0.0100713
## 261 1.8971e-05 586 0.95868 1.01416 0.0100721
## 262 1.8902e-05 587 0.95866 1.01417 0.0100723
## 263 1.8761e-05 590 0.95860 1.01416 0.0100712
## 264 1.8751e-05 591 0.95859 1.01421 0.0100721
## 265 1.8713e-05 593 0.95855 1.01423 0.0100722
## 266 1.8653e-05 595 0.95851 1.01418 0.0100719
## 267 1.8639e-05 598 0.95846 1.01417 0.0100714
## 268 1.8454e-05 599 0.95844 1.01418 0.0100720
## 269 1.8303e-05 601 0.95840 1.01437 0.0100748
## 270 1.8186e-05 602 0.95838 1.01441 0.0100756
## 271 1.8180e-05 603 0.95836 1.01440 0.0100756
## 272 1.8159e-05 604 0.95834 1.01440 0.0100756
## 273 1.8117e-05 605 0.95833 1.01440 0.0100755
## 274 1.7922e-05 606 0.95831 1.01441 0.0100757
## 275 1.7861e-05 608 0.95827 1.01453 0.0100766
## 276 1.7838e-05 609 0.95825 1.01453 0.0100766
## 277 1.7815e-05 610 0.95824 1.01453 0.0100767
## 278 1.7783e-05 611 0.95822 1.01456 0.0100789
## 279 1.7582e-05 614 0.95817 1.01455 0.0100787
## 280 1.7577e-05 615 0.95815 1.01456 0.0100789
## 281 1.7142e-05 618 0.95809 1.01472 0.0100792
## 282 1.7076e-05 620 0.95805 1.01484 0.0100796
## 283 1.7056e-05 621 0.95804 1.01482 0.0100793
## 284 1.6959e-05 622 0.95802 1.01484 0.0100793
## 285 1.6914e-05 625 0.95797 1.01482 0.0100793
## 286 1.6905e-05 628 0.95792 1.01482 0.0100793
## 287 1.6869e-05 629 0.95790 1.01482 0.0100793
## 288 1.6771e-05 630 0.95788 1.01489 0.0100804
## 289 1.6750e-05 631 0.95787 1.01493 0.0100808
## 290 1.6542e-05 632 0.95785 1.01502 0.0100826
## 291 1.6452e-05 634 0.95782 1.01488 0.0100807
## 292 1.6410e-05 635 0.95780 1.01495 0.0100817
## 293 1.6336e-05 636 0.95778 1.01495 0.0100819
## 294 1.6191e-05 637 0.95777 1.01499 0.0100821
## 295 1.6162e-05 638 0.95775 1.01508 0.0100836
## 296 1.6115e-05 640 0.95772 1.01510 0.0100841
## 297 1.6023e-05 641 0.95770 1.01512 0.0100843
## 298 1.5969e-05 642 0.95769 1.01513 0.0100844
## 299 1.5934e-05 643 0.95767 1.01513 0.0100844
## 300 1.5784e-05 644 0.95765 1.01514 0.0100843
## 301 1.5618e-05 645 0.95764 1.01523 0.0100861
## 302 1.5422e-05 647 0.95761 1.01531 0.0100872
## 303 1.5411e-05 648 0.95759 1.01532 0.0100872
## 304 1.5407e-05 650 0.95756 1.01533 0.0100872
## 305 1.5397e-05 651 0.95755 1.01534 0.0100873
## 306 1.5393e-05 653 0.95752 1.01534 0.0100873
## 307 1.5359e-05 656 0.95747 1.01534 0.0100873
## 308 1.5327e-05 657 0.95745 1.01536 0.0100879
## 309 1.5309e-05 658 0.95744 1.01535 0.0100879
## 310 1.5281e-05 661 0.95739 1.01535 0.0100878
## 311 1.5227e-05 662 0.95738 1.01544 0.0100891
## 312 1.5155e-05 663 0.95736 1.01547 0.0100898
## 313 1.5143e-05 666 0.95732 1.01547 0.0100898
## 314 1.5142e-05 672 0.95721 1.01547 0.0100898
## 315 1.5051e-05 673 0.95720 1.01547 0.0100898
## 316 1.4986e-05 674 0.95718 1.01544 0.0100899
## 317 1.4979e-05 676 0.95715 1.01538 0.0100880
## 318 1.4904e-05 677 0.95714 1.01533 0.0100875
## 319 1.4847e-05 678 0.95712 1.01526 0.0100870
## 320 1.4750e-05 680 0.95710 1.01542 0.0100900
## 321 1.4726e-05 681 0.95708 1.01541 0.0100902
## 322 1.4644e-05 682 0.95707 1.01544 0.0100903
## 323 1.4588e-05 684 0.95704 1.01546 0.0100903
## 324 1.4555e-05 685 0.95702 1.01554 0.0100910
## 325 1.4534e-05 686 0.95701 1.01556 0.0100912
## 326 1.4414e-05 687 0.95699 1.01562 0.0100918
## 327 1.4407e-05 688 0.95698 1.01565 0.0100922
## 328 1.4384e-05 690 0.95695 1.01564 0.0100921
## 329 1.4361e-05 691 0.95694 1.01568 0.0100925
## 330 1.4246e-05 693 0.95691 1.01567 0.0100924
## 331 1.4239e-05 696 0.95686 1.01562 0.0100917
## 332 1.4178e-05 697 0.95685 1.01561 0.0100917
## 333 1.4125e-05 699 0.95682 1.01566 0.0100921
## 334 1.3874e-05 700 0.95680 1.01565 0.0100918
## 335 1.3842e-05 701 0.95679 1.01577 0.0100943
## 336 1.3690e-05 702 0.95678 1.01577 0.0100944
## 337 1.3688e-05 703 0.95676 1.01581 0.0100949
## 338 1.3686e-05 704 0.95675 1.01581 0.0100949
## 339 1.3655e-05 705 0.95673 1.01581 0.0100948
## 340 1.3629e-05 707 0.95671 1.01579 0.0100954
## 341 1.3607e-05 708 0.95669 1.01581 0.0100955
## 342 1.3504e-05 709 0.95668 1.01581 0.0100951
## 343 1.3372e-05 710 0.95667 1.01590 0.0100959
## 344 1.3277e-05 711 0.95665 1.01581 0.0100944
## 345 1.3264e-05 712 0.95664 1.01581 0.0100947
## 346 1.3195e-05 713 0.95663 1.01576 0.0100941
## 347 1.3179e-05 714 0.95661 1.01577 0.0100940
## 348 1.3123e-05 715 0.95660 1.01580 0.0100943
## 349 1.3089e-05 716 0.95659 1.01577 0.0100940
## 350 1.3040e-05 717 0.95657 1.01585 0.0100997
## 351 1.2911e-05 718 0.95656 1.01587 0.0100999
## 352 1.2877e-05 719 0.95655 1.01587 0.0101003
## 353 1.2795e-05 721 0.95652 1.01587 0.0101006
## 354 1.2768e-05 722 0.95651 1.01588 0.0101005
## 355 1.2724e-05 726 0.95646 1.01592 0.0101009
## 356 1.2703e-05 727 0.95645 1.01592 0.0101011
## 357 1.2674e-05 728 0.95643 1.01594 0.0101012
## 358 1.2599e-05 729 0.95642 1.01600 0.0101019
## 359 1.2350e-05 730 0.95641 1.01593 0.0101013
## 360 1.2285e-05 731 0.95640 1.01590 0.0101004
## 361 1.2222e-05 732 0.95638 1.01592 0.0101008
## 362 1.2214e-05 733 0.95637 1.01590 0.0101005
## 363 1.2172e-05 734 0.95636 1.01589 0.0101005
## 364 1.2157e-05 736 0.95633 1.01592 0.0101009
## 365 1.2134e-05 737 0.95632 1.01592 0.0101009
## 366 1.2017e-05 738 0.95631 1.01593 0.0101012
## 367 1.1986e-05 739 0.95630 1.01597 0.0101024
## 368 1.1896e-05 740 0.95629 1.01597 0.0101024
## 369 1.1889e-05 741 0.95627 1.01598 0.0101027
## 370 1.1878e-05 742 0.95626 1.01598 0.0101027
## 371 1.1697e-05 744 0.95624 1.01597 0.0101025
## 372 1.1689e-05 746 0.95622 1.01596 0.0101017
## 373 1.1678e-05 747 0.95620 1.01596 0.0101017
## 374 1.1623e-05 748 0.95619 1.01595 0.0101016
## 375 1.1609e-05 749 0.95618 1.01594 0.0101016
## 376 1.1593e-05 750 0.95617 1.01594 0.0101015
## 377 1.1487e-05 751 0.95616 1.01597 0.0101016
## 378 1.1333e-05 752 0.95615 1.01593 0.0101012
## 379 1.1309e-05 753 0.95613 1.01586 0.0101006
## 380 1.1230e-05 755 0.95611 1.01587 0.0101007
## 381 1.1213e-05 756 0.95610 1.01590 0.0101011
## 382 1.1151e-05 757 0.95609 1.01594 0.0101020
## 383 1.1140e-05 760 0.95606 1.01593 0.0101019
## 384 1.1105e-05 761 0.95604 1.01593 0.0101019
## 385 1.1021e-05 762 0.95603 1.01591 0.0101016
## 386 1.1019e-05 765 0.95600 1.01588 0.0101009
## 387 1.0945e-05 766 0.95599 1.01588 0.0101010
## 388 1.0918e-05 767 0.95598 1.01589 0.0101014
## 389 1.0917e-05 768 0.95597 1.01591 0.0101018
## 390 1.0836e-05 771 0.95593 1.01589 0.0101019
## 391 1.0802e-05 773 0.95591 1.01591 0.0101020
## 392 1.0797e-05 774 0.95590 1.01593 0.0101022
## 393 1.0765e-05 775 0.95589 1.01595 0.0101027
## 394 1.0720e-05 776 0.95588 1.01603 0.0101043
## 395 1.0674e-05 777 0.95587 1.01606 0.0101045
## 396 1.0657e-05 778 0.95586 1.01608 0.0101045
## 397 1.0653e-05 779 0.95585 1.01608 0.0101045
## 398 1.0639e-05 780 0.95584 1.01608 0.0101046
## 399 1.0619e-05 781 0.95583 1.01607 0.0101045
## 400 1.0537e-05 782 0.95582 1.01604 0.0101041
## 401 1.0530e-05 783 0.95581 1.01604 0.0101041
## 402 1.0528e-05 784 0.95580 1.01604 0.0101041
## 403 1.0520e-05 785 0.95579 1.01604 0.0101041
## 404 1.0474e-05 786 0.95577 1.01603 0.0101039
## 405 1.0362e-05 787 0.95576 1.01613 0.0101059
## 406 1.0286e-05 788 0.95575 1.01616 0.0101065
## 407 1.0260e-05 789 0.95574 1.01617 0.0101066
## 408 1.0249e-05 790 0.95573 1.01615 0.0101065
## 409 1.0119e-05 791 0.95572 1.01617 0.0101065
## 410 1.0105e-05 792 0.95571 1.01617 0.0101059
## 411 1.0050e-05 793 0.95570 1.01620 0.0101062
## 412 9.8889e-06 794 0.95569 1.01628 0.0101069
## 413 9.8646e-06 795 0.95568 1.01637 0.0101079
## 414 9.8248e-06 802 0.95561 1.01634 0.0101076
## 415 9.8245e-06 803 0.95560 1.01630 0.0101072
## 416 9.8173e-06 806 0.95557 1.01630 0.0101073
## 417 9.8128e-06 808 0.95555 1.01630 0.0101073
## 418 9.7764e-06 809 0.95554 1.01626 0.0101061
## 419 9.7041e-06 810 0.95553 1.01626 0.0101061
## 420 9.6574e-06 812 0.95551 1.01626 0.0101062
## 421 9.6521e-06 813 0.95550 1.01627 0.0101063
## 422 9.5770e-06 814 0.95549 1.01627 0.0101062
## 423 9.5695e-06 815 0.95548 1.01627 0.0101062
## 424 9.5513e-06 816 0.95548 1.01627 0.0101062
## 425 9.4896e-06 817 0.95547 1.01627 0.0101062
## 426 9.4583e-06 821 0.95543 1.01628 0.0101064
## 427 9.3966e-06 822 0.95542 1.01625 0.0101062
## 428 9.3827e-06 825 0.95539 1.01624 0.0101057
## 429 9.3241e-06 828 0.95536 1.01626 0.0101056
## 430 9.1542e-06 829 0.95535 1.01627 0.0101058
## 431 8.9957e-06 831 0.95533 1.01629 0.0101062
## 432 8.9710e-06 834 0.95531 1.01628 0.0101062
## 433 8.8888e-06 836 0.95529 1.01629 0.0101062
## 434 8.8374e-06 838 0.95527 1.01630 0.0101066
## 435 8.8326e-06 839 0.95526 1.01633 0.0101071
## 436 8.5850e-06 840 0.95525 1.01631 0.0101058
## 437 8.5843e-06 841 0.95525 1.01630 0.0101063
## 438 8.5595e-06 842 0.95524 1.01630 0.0101063
## 439 8.5529e-06 843 0.95523 1.01630 0.0101063
## 440 8.5503e-06 844 0.95522 1.01630 0.0101063
## 441 8.5251e-06 846 0.95520 1.01631 0.0101063
## 442 8.4636e-06 847 0.95519 1.01626 0.0101055
## 443 8.4559e-06 848 0.95519 1.01632 0.0101068
## 444 8.4450e-06 849 0.95518 1.01632 0.0101068
## 445 8.3680e-06 851 0.95516 1.01636 0.0101073
## 446 8.3475e-06 852 0.95515 1.01633 0.0101073
## 447 8.3428e-06 853 0.95514 1.01633 0.0101073
## 448 8.3400e-06 854 0.95513 1.01633 0.0101073
## 449 8.3056e-06 856 0.95512 1.01632 0.0101073
## 450 8.1457e-06 857 0.95511 1.01631 0.0101070
## 451 8.1012e-06 859 0.95509 1.01635 0.0101078
## 452 8.0838e-06 860 0.95509 1.01641 0.0101091
## 453 8.0333e-06 861 0.95508 1.01642 0.0101090
## 454 7.9854e-06 864 0.95505 1.01644 0.0101092
## 455 7.9320e-06 865 0.95505 1.01648 0.0101097
## 456 7.8968e-06 866 0.95504 1.01647 0.0101098
## 457 7.8877e-06 868 0.95502 1.01649 0.0101103
## 458 7.7561e-06 869 0.95501 1.01644 0.0101099
## 459 7.6444e-06 870 0.95501 1.01639 0.0101092
## 460 7.5736e-06 871 0.95500 1.01639 0.0101094
## 461 7.5549e-06 873 0.95498 1.01643 0.0101098
## 462 7.5314e-06 874 0.95498 1.01649 0.0101107
## 463 7.5292e-06 875 0.95497 1.01649 0.0101107
## 464 7.5004e-06 876 0.95496 1.01650 0.0101107
## 465 7.3501e-06 877 0.95495 1.01645 0.0101098
## 466 7.3006e-06 878 0.95495 1.01642 0.0101093
## 467 7.2726e-06 880 0.95493 1.01641 0.0101085
## 468 7.2417e-06 881 0.95492 1.01642 0.0101087
## 469 7.1403e-06 882 0.95492 1.01643 0.0101092
## 470 7.1351e-06 884 0.95490 1.01650 0.0101107
## 471 7.1312e-06 885 0.95490 1.01650 0.0101107
## 472 7.1283e-06 886 0.95489 1.01650 0.0101107
## 473 7.0416e-06 887 0.95488 1.01650 0.0101110
## 474 7.0005e-06 890 0.95486 1.01651 0.0101121
## 475 6.9990e-06 891 0.95485 1.01651 0.0101122
## 476 6.9850e-06 892 0.95485 1.01649 0.0101119
## 477 6.8134e-06 894 0.95483 1.01650 0.0101123
## 478 6.8032e-06 895 0.95483 1.01644 0.0101116
## 479 6.7829e-06 896 0.95482 1.01644 0.0101116
## 480 6.7371e-06 900 0.95479 1.01648 0.0101119
## 481 6.7037e-06 901 0.95478 1.01646 0.0101115
## 482 6.6462e-06 902 0.95477 1.01643 0.0101110
## 483 6.6440e-06 903 0.95477 1.01643 0.0101108
## 484 6.5955e-06 904 0.95476 1.01647 0.0101114
## 485 6.5734e-06 905 0.95475 1.01651 0.0101119
## 486 6.5570e-06 906 0.95475 1.01652 0.0101120
## 487 6.5239e-06 908 0.95473 1.01651 0.0101118
## 488 6.5122e-06 909 0.95473 1.01651 0.0101119
## 489 6.4679e-06 911 0.95471 1.01651 0.0101118
## 490 6.4167e-06 912 0.95471 1.01647 0.0101112
## 491 6.4122e-06 913 0.95470 1.01645 0.0101110
## 492 6.3689e-06 914 0.95469 1.01644 0.0101106
## 493 6.3510e-06 915 0.95469 1.01642 0.0101105
## 494 6.3308e-06 917 0.95468 1.01641 0.0101104
## 495 6.3188e-06 918 0.95467 1.01641 0.0101106
## 496 6.3175e-06 919 0.95466 1.01640 0.0101105
## 497 6.2880e-06 920 0.95466 1.01640 0.0101104
## 498 5.9459e-06 921 0.95465 1.01646 0.0101110
## 499 5.8722e-06 922 0.95464 1.01659 0.0101129
## 500 5.8368e-06 924 0.95463 1.01661 0.0101135
## 501 5.8292e-06 925 0.95463 1.01661 0.0101136
## 502 5.7827e-06 926 0.95462 1.01659 0.0101133
## 503 5.6506e-06 927 0.95462 1.01660 0.0101129
## 504 5.6071e-06 928 0.95461 1.01659 0.0101128
## 505 5.5875e-06 929 0.95460 1.01659 0.0101128
## 506 5.4194e-06 930 0.95460 1.01658 0.0101120
## 507 5.4030e-06 931 0.95459 1.01656 0.0101115
## 508 5.3822e-06 932 0.95459 1.01655 0.0101114
## 509 5.3769e-06 935 0.95457 1.01656 0.0101114
## 510 5.3552e-06 936 0.95457 1.01656 0.0101114
## 511 5.3459e-06 937 0.95456 1.01655 0.0101111
## 512 5.2851e-06 938 0.95456 1.01654 0.0101105
## 513 5.1883e-06 939 0.95455 1.01648 0.0101103
## 514 5.1687e-06 941 0.95454 1.01648 0.0101099
## 515 5.1249e-06 942 0.95453 1.01649 0.0101101
## 516 5.1045e-06 943 0.95453 1.01637 0.0101073
## 517 5.1037e-06 944 0.95452 1.01638 0.0101074
## 518 5.0790e-06 945 0.95452 1.01638 0.0101074
## 519 5.0133e-06 946 0.95451 1.01643 0.0101077
## 520 4.9829e-06 947 0.95451 1.01642 0.0101078
## 521 4.7069e-06 950 0.95449 1.01644 0.0101078
## 522 4.7052e-06 951 0.95449 1.01646 0.0101080
## 523 4.6384e-06 953 0.95448 1.01650 0.0101083
## 524 4.5861e-06 954 0.95447 1.01649 0.0101083
## 525 4.5321e-06 955 0.95447 1.01653 0.0101086
## 526 4.5266e-06 956 0.95446 1.01654 0.0101085
## 527 4.5201e-06 957 0.95446 1.01654 0.0101085
## 528 4.5197e-06 958 0.95445 1.01654 0.0101085
## 529 4.4636e-06 959 0.95445 1.01661 0.0101096
## 530 4.4113e-06 960 0.95445 1.01664 0.0101097
## 531 4.3985e-06 961 0.95444 1.01664 0.0101097
## 532 4.3898e-06 962 0.95444 1.01663 0.0101096
## 533 4.3863e-06 963 0.95443 1.01662 0.0101095
## 534 4.3670e-06 965 0.95442 1.01662 0.0101094
## 535 4.3536e-06 967 0.95442 1.01663 0.0101095
## 536 4.2226e-06 968 0.95441 1.01660 0.0101097
## 537 4.1748e-06 971 0.95440 1.01661 0.0101098
## 538 4.0876e-06 973 0.95439 1.01658 0.0101096
## 539 4.0873e-06 974 0.95439 1.01659 0.0101098
## 540 4.0419e-06 975 0.95438 1.01658 0.0101098
## 541 4.0147e-06 976 0.95438 1.01661 0.0101102
## 542 3.9934e-06 977 0.95437 1.01663 0.0101104
## 543 3.9915e-06 978 0.95437 1.01665 0.0101105
## 544 3.9458e-06 979 0.95437 1.01663 0.0101102
## 545 3.9088e-06 980 0.95436 1.01665 0.0101103
## 546 3.8950e-06 981 0.95436 1.01667 0.0101107
## 547 3.8520e-06 982 0.95435 1.01667 0.0101107
## 548 3.8325e-06 983 0.95435 1.01674 0.0101113
## 549 3.8217e-06 985 0.95434 1.01675 0.0101113
## 550 3.8038e-06 986 0.95434 1.01674 0.0101112
## 551 3.7757e-06 987 0.95433 1.01674 0.0101112
## 552 3.7173e-06 989 0.95433 1.01675 0.0101113
## 553 3.7151e-06 991 0.95432 1.01676 0.0101113
## 554 3.7051e-06 993 0.95431 1.01676 0.0101114
## 555 3.6763e-06 996 0.95430 1.01677 0.0101116
## 556 3.6713e-06 999 0.95429 1.01676 0.0101118
## 557 3.6514e-06 1000 0.95429 1.01675 0.0101116
## 558 3.6479e-06 1001 0.95428 1.01675 0.0101116
## 559 3.6312e-06 1002 0.95428 1.01675 0.0101116
## 560 3.6267e-06 1005 0.95427 1.01675 0.0101116
## 561 3.5636e-06 1006 0.95426 1.01673 0.0101115
## 562 3.5612e-06 1007 0.95426 1.01671 0.0101115
## 563 3.5338e-06 1008 0.95426 1.01672 0.0101116
## 564 3.4795e-06 1009 0.95425 1.01672 0.0101117
## 565 3.4271e-06 1010 0.95425 1.01663 0.0101106
## 566 3.4180e-06 1011 0.95425 1.01666 0.0101109
## 567 3.4043e-06 1013 0.95424 1.01666 0.0101109
## 568 3.3814e-06 1014 0.95424 1.01666 0.0101109
## 569 3.3741e-06 1015 0.95423 1.01667 0.0101109
## 570 3.3659e-06 1016 0.95423 1.01667 0.0101109
## 571 3.3621e-06 1017 0.95423 1.01669 0.0101112
## 572 3.2979e-06 1018 0.95422 1.01669 0.0101111
## 573 3.2375e-06 1019 0.95422 1.01670 0.0101115
## 574 3.2154e-06 1020 0.95422 1.01669 0.0101103
## 575 3.2118e-06 1021 0.95421 1.01669 0.0101101
## 576 3.2029e-06 1022 0.95421 1.01669 0.0101101
## 577 3.1599e-06 1023 0.95421 1.01667 0.0101100
## 578 3.1478e-06 1025 0.95420 1.01669 0.0101104
## 579 3.1055e-06 1026 0.95420 1.01674 0.0101110
## 580 3.1052e-06 1027 0.95419 1.01675 0.0101111
## 581 3.1044e-06 1029 0.95419 1.01675 0.0101111
## 582 3.0057e-06 1030 0.95419 1.01678 0.0101114
## 583 2.9856e-06 1031 0.95418 1.01679 0.0101119
## 584 2.9656e-06 1032 0.95418 1.01679 0.0101120
## 585 2.8887e-06 1033 0.95418 1.01675 0.0101113
## 586 2.8408e-06 1034 0.95417 1.01674 0.0101109
## 587 2.8350e-06 1035 0.95417 1.01674 0.0101109
## 588 2.8145e-06 1037 0.95416 1.01673 0.0101110
## 589 2.7021e-06 1039 0.95416 1.01668 0.0101100
## 590 2.6975e-06 1040 0.95416 1.01665 0.0101094
## 591 2.6415e-06 1041 0.95415 1.01666 0.0101095
## 592 2.6173e-06 1042 0.95415 1.01667 0.0101096
## 593 2.5785e-06 1043 0.95415 1.01668 0.0101096
## 594 2.5634e-06 1044 0.95415 1.01670 0.0101099
## 595 2.5555e-06 1045 0.95414 1.01670 0.0101099
## 596 2.4943e-06 1046 0.95414 1.01670 0.0101099
## 597 2.4582e-06 1047 0.95414 1.01668 0.0101098
## 598 2.4480e-06 1048 0.95414 1.01669 0.0101098
## 599 2.3943e-06 1049 0.95413 1.01669 0.0101098
## 600 2.3745e-06 1050 0.95413 1.01670 0.0101099
## 601 2.2457e-06 1051 0.95413 1.01667 0.0101094
## 602 2.2251e-06 1052 0.95413 1.01669 0.0101096
## 603 2.2158e-06 1054 0.95412 1.01671 0.0101098
## 604 2.2033e-06 1055 0.95412 1.01671 0.0101098
## 605 2.1392e-06 1056 0.95412 1.01672 0.0101102
## 606 2.1150e-06 1057 0.95412 1.01672 0.0101096
## 607 2.0265e-06 1058 0.95411 1.01676 0.0101100
## 608 2.0221e-06 1059 0.95411 1.01677 0.0101098
## 609 1.9878e-06 1060 0.95411 1.01676 0.0101096
## 610 1.9843e-06 1061 0.95411 1.01675 0.0101094
## 611 1.9780e-06 1062 0.95411 1.01674 0.0101093
## 612 1.9639e-06 1063 0.95410 1.01675 0.0101094
## 613 1.9217e-06 1064 0.95410 1.01677 0.0101095
## 614 1.8899e-06 1065 0.95410 1.01676 0.0101097
## 615 1.8514e-06 1066 0.95410 1.01666 0.0101091
## 616 1.8477e-06 1067 0.95410 1.01662 0.0101083
## 617 1.8276e-06 1068 0.95409 1.01663 0.0101084
## 618 1.8122e-06 1069 0.95409 1.01664 0.0101085
## 619 1.8060e-06 1070 0.95409 1.01663 0.0101085
## 620 1.7232e-06 1071 0.95409 1.01666 0.0101088
## 621 1.7188e-06 1072 0.95409 1.01664 0.0101086
## 622 1.6816e-06 1073 0.95408 1.01660 0.0101084
## 623 1.6167e-06 1074 0.95408 1.01657 0.0101081
## 624 1.6037e-06 1075 0.95408 1.01659 0.0101084
## 625 1.5854e-06 1077 0.95408 1.01658 0.0101084
## 626 1.5740e-06 1078 0.95408 1.01659 0.0101085
## 627 1.5678e-06 1079 0.95408 1.01660 0.0101086
## 628 1.5542e-06 1080 0.95407 1.01660 0.0101085
## 629 1.5300e-06 1081 0.95407 1.01658 0.0101083
## 630 1.5144e-06 1082 0.95407 1.01657 0.0101084
## 631 1.4835e-06 1083 0.95407 1.01656 0.0101084
## 632 1.4507e-06 1084 0.95407 1.01658 0.0101087
## 633 1.4435e-06 1085 0.95407 1.01656 0.0101086
## 634 1.4159e-06 1086 0.95406 1.01657 0.0101087
## 635 1.3936e-06 1088 0.95406 1.01656 0.0101086
## 636 1.3452e-06 1089 0.95406 1.01657 0.0101088
## 637 1.3354e-06 1090 0.95406 1.01658 0.0101087
## 638 1.3301e-06 1091 0.95406 1.01657 0.0101087
## 639 1.2813e-06 1092 0.95406 1.01647 0.0101073
## 640 1.2740e-06 1093 0.95406 1.01646 0.0101072
## 641 1.2118e-06 1094 0.95405 1.01645 0.0101070
## 642 1.2025e-06 1095 0.95405 1.01646 0.0101073
## 643 1.1985e-06 1096 0.95405 1.01646 0.0101074
## 644 1.1837e-06 1097 0.95405 1.01646 0.0101073
## 645 1.1739e-06 1098 0.95405 1.01646 0.0101074
## 646 1.1189e-06 1099 0.95405 1.01645 0.0101072
## 647 1.1003e-06 1101 0.95405 1.01645 0.0101077
## 648 1.0877e-06 1102 0.95404 1.01646 0.0101079
## 649 1.0866e-06 1103 0.95404 1.01645 0.0101079
## 650 1.0865e-06 1105 0.95404 1.01645 0.0101079
## 651 1.0789e-06 1106 0.95404 1.01645 0.0101079
## 652 1.0593e-06 1107 0.95404 1.01645 0.0101079
## 653 9.8972e-07 1108 0.95404 1.01647 0.0101079
## 654 9.5559e-07 1111 0.95403 1.01646 0.0101076
## 655 9.2998e-07 1112 0.95403 1.01644 0.0101070
## 656 8.6389e-07 1113 0.95403 1.01643 0.0101069
## 657 8.4989e-07 1114 0.95403 1.01640 0.0101066
## 658 8.3215e-07 1115 0.95403 1.01639 0.0101063
## 659 8.2266e-07 1116 0.95403 1.01639 0.0101063
## 660 7.6430e-07 1117 0.95403 1.01637 0.0101060
## 661 7.6326e-07 1118 0.95403 1.01637 0.0101060
## 662 7.6324e-07 1119 0.95403 1.01637 0.0101060
## 663 7.6122e-07 1120 0.95403 1.01637 0.0101060
## 664 7.5885e-07 1121 0.95403 1.01637 0.0101059
## 665 7.5877e-07 1123 0.95403 1.01637 0.0101059
## 666 7.3962e-07 1124 0.95402 1.01634 0.0101056
## 667 7.3351e-07 1125 0.95402 1.01636 0.0101057
## 668 7.1373e-07 1126 0.95402 1.01636 0.0101057
## 669 7.1191e-07 1127 0.95402 1.01636 0.0101058
## 670 7.0841e-07 1128 0.95402 1.01636 0.0101059
## 671 6.7300e-07 1129 0.95402 1.01633 0.0101056
## 672 6.5995e-07 1130 0.95402 1.01635 0.0101059
## 673 6.5757e-07 1131 0.95402 1.01634 0.0101059
## 674 6.4036e-07 1132 0.95402 1.01635 0.0101060
## 675 6.3530e-07 1133 0.95402 1.01634 0.0101059
## 676 6.1866e-07 1134 0.95402 1.01634 0.0101059
## 677 6.0943e-07 1135 0.95402 1.01635 0.0101060
## 678 5.9369e-07 1136 0.95402 1.01636 0.0101062
## 679 5.9312e-07 1137 0.95402 1.01636 0.0101062
## 680 5.7425e-07 1138 0.95402 1.01635 0.0101061
## 681 5.7424e-07 1139 0.95401 1.01635 0.0101060
## 682 5.5930e-07 1140 0.95401 1.01634 0.0101058
## 683 5.2954e-07 1141 0.95401 1.01634 0.0101058
## 684 5.0927e-07 1142 0.95401 1.01635 0.0101060
## 685 4.8484e-07 1143 0.95401 1.01637 0.0101061
## 686 4.7725e-07 1144 0.95401 1.01637 0.0101062
## 687 4.7156e-07 1145 0.95401 1.01637 0.0101061
## 688 4.6285e-07 1146 0.95401 1.01637 0.0101061
## 689 4.6100e-07 1147 0.95401 1.01638 0.0101063
## 690 4.5842e-07 1148 0.95401 1.01637 0.0101063
## 691 4.5502e-07 1149 0.95401 1.01638 0.0101063
## 692 4.4877e-07 1150 0.95401 1.01638 0.0101063
## 693 4.4038e-07 1151 0.95401 1.01638 0.0101063
## 694 4.3712e-07 1152 0.95401 1.01639 0.0101063
## 695 4.1946e-07 1153 0.95401 1.01638 0.0101062
## 696 4.1814e-07 1154 0.95401 1.01639 0.0101063
## 697 4.1721e-07 1156 0.95401 1.01639 0.0101063
## 698 4.1522e-07 1157 0.95401 1.01639 0.0101063
## 699 3.9583e-07 1158 0.95401 1.01639 0.0101063
## 700 3.7078e-07 1159 0.95401 1.01640 0.0101064
## 701 3.6954e-07 1160 0.95400 1.01642 0.0101066
## 702 3.6058e-07 1161 0.95400 1.01642 0.0101065
## 703 3.1961e-07 1163 0.95400 1.01641 0.0101064
## 704 3.1492e-07 1164 0.95400 1.01641 0.0101065
## 705 2.7853e-07 1165 0.95400 1.01640 0.0101065
## 706 2.5540e-07 1166 0.95400 1.01639 0.0101063
## 707 2.5514e-07 1167 0.95400 1.01638 0.0101062
## 708 2.4123e-07 1168 0.95400 1.01639 0.0101063
## 709 2.3869e-07 1169 0.95400 1.01639 0.0101063
## 710 2.3797e-07 1170 0.95400 1.01639 0.0101063
## 711 2.2586e-07 1171 0.95400 1.01639 0.0101063
## 712 2.2166e-07 1172 0.95400 1.01639 0.0101064
## 713 1.9000e-07 1173 0.95400 1.01639 0.0101064
## 714 1.5415e-07 1174 0.95400 1.01640 0.0101067
## 715 1.5085e-07 1175 0.95400 1.01641 0.0101067
## 716 1.4196e-07 1176 0.95400 1.01641 0.0101067
## 717 1.3557e-07 1178 0.95400 1.01638 0.0101056
## 718 1.3414e-07 1179 0.95400 1.01637 0.0101055
## 719 1.2495e-07 1180 0.95400 1.01636 0.0101054
## 720 1.1478e-07 1182 0.95400 1.01636 0.0101054
## 721 1.0872e-07 1184 0.95400 1.01636 0.0101054
## 722 1.0314e-07 1185 0.95400 1.01636 0.0101053
## 723 9.2557e-08 1186 0.95400 1.01636 0.0101053
## 724 8.6474e-08 1188 0.95400 1.01636 0.0101052
## 725 8.3211e-08 1189 0.95400 1.01636 0.0101053
## 726 7.9607e-08 1190 0.95400 1.01636 0.0101053
## 727 7.9471e-08 1191 0.95400 1.01637 0.0101053
## 728 7.5845e-08 1192 0.95400 1.01637 0.0101053
## 729 5.8589e-08 1193 0.95400 1.01637 0.0101053
## 730 5.6947e-08 1194 0.95400 1.01637 0.0101053
## 731 4.4922e-08 1195 0.95400 1.01637 0.0101053
## 732 4.2453e-08 1196 0.95400 1.01637 0.0101053
## 733 4.1586e-08 1197 0.95400 1.01637 0.0101053
## 734 2.6870e-08 1198 0.95400 1.01637 0.0101053
## 735 2.6263e-08 1199 0.95400 1.01635 0.0101051
## 736 2.4402e-08 1201 0.95400 1.01635 0.0101051
## 737 2.1248e-08 1202 0.95400 1.01635 0.0101050
## 738 1.7800e-08 1203 0.95400 1.01635 0.0101050
## 739 1.6096e-08 1204 0.95400 1.01635 0.0101050
## 740 1.5832e-08 1205 0.95400 1.01635 0.0101050
## 741 5.6681e-09 1207 0.95400 1.01634 0.0101050
## 742 5.4882e-09 1208 0.95400 1.01634 0.0101050
## 743 4.6971e-09 1209 0.95400 1.01634 0.0101050
## 744 2.8927e-09 1210 0.95400 1.01633 0.0101048
## 745 1.2189e-09 1211 0.95400 1.01633 0.0101048
## 746 0.0000e+00 1212 0.95400 1.01633 0.0101048We extract the optimal complexity parameter.
plot(m0_rpart$cptable[,c(1,4)], xlim=c(0,0.001))
cp_star = m0_rpart$cptable[which.min(m0_rpart$cptable[,4]),1]Let us see the optimal tree.
rpart.plot(prune(m0_rpart,cp=cp_star))
2.3 All covariates
Let us now include all the covariates.
m1_rpart = rpart(cbind(Exposure, ClaimNb)~ Power+ CarAge+DriverAge+Brand+Gas+Region+Density,
data=training_set,
method="poisson",
control = rpart.control(cp = 0, xval = 10, minbucket = 1000))
printcp(m1_rpart)##
## Rates regression tree:
## rpart(formula = cbind(Exposure, ClaimNb) ~ Power + CarAge + DriverAge +
## Brand + Gas + Region + Density, data = training_set, method = "poisson",
## control = rpart.control(cp = 0, xval = 10, minbucket = 1000))
##
## Variables actually used in tree construction:
## [1] Brand CarAge Density DriverAge Gas Power Region
##
## Root node error: 42075/164346 = 0.25602
##
## n= 164346
##
## CP nsplit rel error xerror xstd
## 1 7.2911e-03 0 1.00000 1.00004 0.0097609
## 2 4.5597e-03 1 0.99271 0.99357 0.0096725
## 3 1.7755e-03 2 0.98815 0.98897 0.0096101
## 4 7.3360e-04 3 0.98637 0.98764 0.0096010
## 5 6.1473e-04 4 0.98564 0.98755 0.0096122
## 6 5.9205e-04 5 0.98503 0.98758 0.0096151
## 7 5.5886e-04 6 0.98443 0.98733 0.0096119
## 8 5.4769e-04 7 0.98387 0.98741 0.0096145
## 9 4.7538e-04 8 0.98333 0.98705 0.0096148
## 10 3.9095e-04 9 0.98285 0.98694 0.0096171
## 11 3.6769e-04 10 0.98246 0.98724 0.0096263
## 12 3.0787e-04 11 0.98209 0.98737 0.0096299
## 13 2.9261e-04 13 0.98148 0.98701 0.0096339
## 14 2.7230e-04 14 0.98118 0.98744 0.0096431
## 15 2.6851e-04 15 0.98091 0.98753 0.0096459
## 16 2.6453e-04 16 0.98064 0.98757 0.0096469
## 17 2.5844e-04 17 0.98038 0.98782 0.0096498
## 18 2.5688e-04 19 0.97986 0.98790 0.0096505
## 19 2.5393e-04 20 0.97961 0.98800 0.0096513
## 20 2.5362e-04 22 0.97910 0.98804 0.0096519
## 21 2.3820e-04 23 0.97884 0.98828 0.0096550
## 22 2.3469e-04 24 0.97861 0.98842 0.0096626
## 23 2.2816e-04 26 0.97814 0.98832 0.0096628
## 24 2.2258e-04 28 0.97768 0.98825 0.0096614
## 25 2.1524e-04 29 0.97746 0.98839 0.0096647
## 26 2.1444e-04 30 0.97724 0.98858 0.0096646
## 27 2.0262e-04 31 0.97703 0.98850 0.0096669
## 28 2.0025e-04 32 0.97683 0.98909 0.0096768
## 29 1.8958e-04 33 0.97662 0.98934 0.0096825
## 30 1.7733e-04 34 0.97644 0.98951 0.0096878
## 31 1.7464e-04 35 0.97626 0.98957 0.0096915
## 32 1.6918e-04 36 0.97608 0.98954 0.0096913
## 33 1.5903e-04 37 0.97591 0.98994 0.0096964
## 34 1.5444e-04 39 0.97560 0.99035 0.0097027
## 35 1.5392e-04 41 0.97529 0.99044 0.0097039
## 36 1.4565e-04 42 0.97513 0.99040 0.0097046
## 37 1.3900e-04 43 0.97499 0.99076 0.0097097
## 38 1.3554e-04 44 0.97485 0.99086 0.0097103
## 39 1.3163e-04 46 0.97458 0.99094 0.0097121
## 40 1.3111e-04 47 0.97445 0.99095 0.0097125
## 41 1.3049e-04 48 0.97431 0.99098 0.0097127
## 42 1.2864e-04 50 0.97405 0.99098 0.0097127
## 43 1.2282e-04 51 0.97393 0.99086 0.0097153
## 44 1.2118e-04 52 0.97380 0.99081 0.0097161
## 45 1.1691e-04 53 0.97368 0.99095 0.0097184
## 46 1.1605e-04 54 0.97356 0.99112 0.0097207
## 47 1.1358e-04 55 0.97345 0.99124 0.0097224
## 48 1.1021e-04 56 0.97333 0.99104 0.0097203
## 49 1.1012e-04 57 0.97322 0.99101 0.0097201
## 50 1.0580e-04 59 0.97300 0.99102 0.0097189
## 51 1.0507e-04 60 0.97290 0.99103 0.0097194
## 52 1.0375e-04 61 0.97279 0.99103 0.0097194
## 53 1.0286e-04 62 0.97269 0.99105 0.0097194
## 54 1.0178e-04 63 0.97259 0.99103 0.0097197
## 55 1.0090e-04 64 0.97248 0.99098 0.0097191
## 56 9.6719e-05 65 0.97238 0.99103 0.0097201
## 57 9.3258e-05 67 0.97219 0.99116 0.0097219
## 58 9.1683e-05 68 0.97210 0.99128 0.0097245
## 59 9.1587e-05 69 0.97201 0.99133 0.0097249
## 60 9.0038e-05 70 0.97191 0.99140 0.0097257
## 61 8.2881e-05 72 0.97173 0.99144 0.0097282
## 62 7.9999e-05 75 0.97149 0.99149 0.0097291
## 63 7.7731e-05 76 0.97141 0.99150 0.0097300
## 64 7.4140e-05 78 0.97125 0.99147 0.0097302
## 65 7.2613e-05 79 0.97118 0.99144 0.0097294
## 66 6.9146e-05 80 0.97110 0.99144 0.0097304
## 67 6.8256e-05 82 0.97096 0.99146 0.0097307
## 68 6.4365e-05 83 0.97090 0.99168 0.0097347
## 69 5.2552e-05 84 0.97083 0.99177 0.0097365
## 70 4.9960e-05 86 0.97073 0.99179 0.0097368
## 71 4.7599e-05 88 0.97063 0.99181 0.0097371
## 72 0.0000e+00 89 0.97058 0.99182 0.0097380We can plot the errors
require(ggplot2)
plotcp(x = m1_rpart,minline = TRUE, col="red")
ggplot() + geom_line(aes(x = m1_rpart$cptable[,1], y=m1_rpart$cptable[,4]))
If we take the value of cp that minimizes the error, we find
cp_star = m1_rpart$cptable[which.min(m1_rpart$cptable[,4]),1]
cp_star## [1] 0.0003909519Let us plot the optimal tree
m2_rpart = prune(m1_rpart, cp=cp_star)
rpart.plot(m2_rpart)
Finally, let us compute the deviance on the testing_set.
2*(sum(dpois(x = testing_set$ClaimNb, lambda = testing_set$ClaimNb,log=TRUE))-
sum(dpois(x = testing_set$ClaimNb, lambda = predict(m2_rpart,testing_set)*testing_set$Exposure,
log=TRUE)))## [1] 10304.12If we compute the deviance on the full tree (not the pruned tree), we obtain
2*(sum(dpois(x = testing_set$ClaimNb, lambda = testing_set$ClaimNb,log=TRUE))-
sum(dpois(x = testing_set$ClaimNb, lambda = predict(m1_rpart,testing_set)*testing_set$Exposure,
log=TRUE)))## [1] 10353.513 Bagging of trees
Let us create the bootstrap samples.
set.seed(85)
bootstrap_samples = createResample(training_set$ClaimNb, times=50)For each sample, we estimate a CART with the optimal complexity parameter found previously. Each tree, gives us an estimation of the claim frequency, which we average.
bagg_cart = lapply(bootstrap_samples, function(X){
rpart(cbind(Exposure, ClaimNb)~ Power+ CarAge+DriverAge+Brand+Gas+Region+Density,
data=training_set[X,],
method="poisson",
control = rpart.control(cp = cp_star, xval = 0))
})pred = lapply(bagg_cart, function(X){
predict(X,testing_set)*testing_set$Exposure})
pred = do.call(cbind, pred)
pred = apply(pred, 1, mean)2*(sum(dpois(x = testing_set$ClaimNb, lambda = testing_set$ClaimNb,log=TRUE))-
sum(dpois(x = testing_set$ClaimNb, lambda = pred,
log=TRUE)))## [1] 10258.92